Simultaneity on the Rotating Disk

Abstract

The disk that rotates in an inertial frame in special relativity has long been analysed by assuming a Lorentz contraction of its peripheral elements in that frame, which has produced widely varying views in the literature. We show that this assumption is unnecessary for a disk that corresponds to the simplest form of rotation in special relativity. After constructing such a disk and showing that observers at rest on it do not constitute a true rotating frame, we choose a “master” observer and calculate a set of disk coordinates and spacetime metric pertinent to that observer. We use this formalism to resolve the “circular twin paradox”, then calculate the speed of light sent around the periphery as measured by the master observer, to show that this speed is a function of sent-direction and disk angle traversed. This result is consistent with the Sagnac Effect, but constitutes a finer analysis of that effect, which is normally expressed using an average speed for a full trip of the periphery. We also use the formalism to give a resolution of “Selleri’s paradox”.

Appendix: Derivation of the Spacetime Metric

In this appendix we derive (20). We require to calculate the elements $g_{\alpha ^{\prime }\beta ^{\prime }}$ of the primed metric in (20), given the elements $g_{\mu \nu }$ of the lab (unprimed) metric in (18). Write the metric tensor as ${\mathsf {g}},$ its matrix of elements with respect to primed coordinates as $[{\mathsf {g}}]_\text {p},$ its matrix of elements with respect to unprimed coordinates as $[{\mathsf {g}}]_\text {u},$ the jacobian matrix of partial derivatives of unprimed coordinates with respect to primed coordinates as $\varLambda ^{\!\text {u}}_\text {p},$ and its inverse, the jacobian matrix of partial derivatives of primed coordinates with respect to unprimed, as $\varLambda ^{\!\text {p}}_\text {u}.$ Then the change-of-variables identity for the metric,

$$\begin{aligned} g_{\alpha ^{\prime }\beta ^{\prime }} = \varLambda ^\mu _{\alpha ^{\prime }} \varLambda ^\nu _{\beta ^{\prime }} g_{\mu \nu }\,, \end{aligned}$$

(49)

can be written in matrix form as (with superscript “t” denoting transpose)

$$\begin{aligned}{}[{\mathsf {g}}]_\text {p} = \big (\varLambda ^{\!\text {u}}_\text {p}\big )^{\text {t}} [{\mathsf {g}}]_\text {u} \varLambda ^{\!\text {u}}_\text {p}. \end{aligned}$$

(50)

Rather than calculate $\varLambda ^{\!\text {u}}_\text {p}$ directly, it is easier to calculate $\varLambda ^{\!\text {p}}_\text {u}$ and invert it. So consider

$$\begin{aligned} \varLambda ^{\!\text {p}}_\text {u}\equiv \begin{bmatrix} \,\partial t^{\prime }/\partial t&\;\;\; \partial t^{\prime }/\partial r&\;\;\; \partial t^{\prime }/\partial \theta \,\\[1.5ex] \,\partial r^{\prime }/\partial t&\;\;\; \partial r^{\prime }/\partial r&\;\;\; \partial r^{\prime }/\partial \theta \,\\[1.5ex] \,\partial \theta ^{\prime }/\partial t&\;\;\; \partial \theta ^{\prime }/\partial r&\;\;\; \partial \theta ^{\prime }/\partial \theta \,\end{bmatrix}. \end{aligned}$$

(51)

Refer to (13) for all coordinate dependencies. We will calculate the first element of the matrix, as this is completely representative of the other elements. Write

$$\begin{aligned} {\partial t^{\prime }\over \partial t} = {1\over \gamma }{\partial t_0\over \partial t}\quad \text {at constant}\,r,\,\theta . \end{aligned}$$

(52)

Use the “$S,\,C\,$” shorthand of (19), and apply $\partial /\partial t$ at constant r and $\theta$ to (10), giving

$$\begin{aligned} {\partial t_0\over \partial t} = 1 + Vr C\varOmega {\partial t_0\over \partial t}, \end{aligned}$$

(53)

which solves for $\partial t_0/\partial t,$ leading to

$$\begin{aligned} {\partial t^{\prime }\over \partial t} = {1\over \gamma (1-Vr C\varOmega )}. \end{aligned}$$

(54)

The other elements of (51) are calculated in the same way; for the bottom row, refer also to the expression for $\theta ^{\prime }$ in (13). We finally obtain

$$\begin{aligned}&\varLambda ^{\!\text {p}}_\text {u}= \begin{bmatrix} \,1\over \gamma (1-Vr C\varOmega )&\;\;\; -VS\over \gamma (1-Vr C\varOmega )&\;\;\; -Vr C\over \gamma (1-Vr C\varOmega )\,\\[1.5ex] \,0&\;\;\; 1&\;\;\; 0\,\\[1.5ex] \,-\varOmega&\;\;\; 0&\;\;\; 1\,\end{bmatrix},\nonumber \\&\quad \text { which inverts to}\, \varLambda ^{\!\text {u}}_\text {p}= \begin{bmatrix} \,\gamma&\;\;\; VS\over 1-Vr C\varOmega&\;\;\; Vr C\over 1-Vr C\varOmega \,\\[1.5ex] \,0&\;\;\; 1&\;\;\; 0\,\\[1.5ex] \,\gamma \varOmega&\;\;\; VS\varOmega \over 1-Vr C\varOmega&\;\;\; 1\over 1-Vr C\varOmega \,\end{bmatrix}. \end{aligned}$$

(55)

We can now evaluate (50). Using the flat-spacetime metric $[{\mathsf {g}}]_\text {u} = {{\mathrm{diag}}}(1,\,-1,\,-r^2)$, the primed-coordinates metric is

$$\begin{aligned}{}[{\mathsf {g}}]_\text {p} = \begin{bmatrix} \,\gamma ^2(1-\varOmega ^2 r^2)&\;\;\; \gamma V S (1-\varOmega ^2 r^2)\over 1 -Vr C\varOmega&\;\;\; \gamma r(VC-\varOmega r)\over 1 -Vr C\varOmega \,\\[2ex] \,\gamma V S (1-\varOmega ^2 r^2)\over 1 -Vr C\varOmega&\;\;\; -1+{V^2S^2(1-\varOmega ^2 r^2)\over (1-Vr C\varOmega )^2}&\;\;\; VSr (VC-\varOmega r)\over (1-Vr C\varOmega )^2\,\\[2ex] \,\gamma r(VC-\varOmega r)\over 1 -VrC\varOmega&\;\;\; VSr(VC-\varOmega r)\over (1-VrC\varOmega )^2&\;\;\; {-r^2(1-V^2C^2)\over (1-VrC\varOmega )^2}\,\end{bmatrix}. \end{aligned}$$

(56)

Now invoke (13) to rewrite r as $r^{\prime },$ after which this metric is easily written with infinitesimals as (20).